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Mirrors > Home > MPE Home > Th. List > 3orbi123d | Structured version Visualization version GIF version |
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
Ref | Expression |
---|---|
bi3d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bi3d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
bi3d.3 | ⊢ (𝜑 → (𝜂 ↔ 𝜁)) |
Ref | Expression |
---|---|
3orbi123d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | bi3d.2 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
3 | 1, 2 | orbi12d 916 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
4 | bi3d.3 | . . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁)) | |
5 | 3, 4 | orbi12d 916 | . 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁))) |
6 | df-3or 1085 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
7 | df-3or 1085 | . 2 ⊢ ((𝜒 ∨ 𝜏 ∨ 𝜁) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁)) | |
8 | 5, 6, 7 | 3bitr4g 317 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 844 ∨ w3o 1083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-or 845 df-3or 1085 |
This theorem is referenced by: moeq3 3651 soeq1 5458 solin 5462 soinxp 5597 ordtri3or 6191 isosolem 7079 sorpssi 7435 dfwe2 7476 f1oweALT 7655 soxp 7806 elfiun 8878 sornom 9688 ltsopr 10443 elz 11971 dyaddisj 24200 istrkgl 26252 istrkgld 26253 axtgupdim2 26265 tgdim01 26301 tglngval 26345 tgellng 26347 colcom 26352 colrot1 26353 legso 26393 lncom 26416 lnrot1 26417 lnrot2 26418 ttgval 26669 colinearalg 26704 axlowdim2 26754 axlowdim 26755 elntg 26778 elntg2 26779 nb3grprlem2 27171 frgrwopreg 28108 istrkg2d 32047 axtgupdim2ALTV 32049 brcolinear2 33632 colineardim1 33635 colinearperm1 33636 fin2so 35044 uneqsn 40726 3orbi123 41217 |
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